Using the provided Venn diagram and definitions, identify equivalent sets and calculate the square root of the given numbers using prime factorization.

Step 1: Solve Question 1. Definition of a proper subset:* A set A is a proper subset of a set B if all elements of A are also elements of B, and B contains at least one element not in A. This is denoted as A B. Elements of sets A, B, and C from the diagram:* A = \b, c, d, e, f\ B = \d, e, f, g\ C = \f, g, h, i\ Identify equivalent sets:* Two sets are equivalent if they have the same number of elements (cardinality). Number of elements in A, |A| = 5. Number of elements in B, |B| = 4. Number of elements in C, |C| = 4. Since |B| = |C|, sets B and C are equivalent sets. The equivalent sets are B and C. Step 2: Solve Question 2 by finding the square root using prime factorization. a) For 50176:* 50176 = 2 × 25088 = 2 × 2 × 12544 = 2 × 2 × 2 × 6272 = 2 × 2 × 2 × 2 × 3136 = 2 × 2 × 2 × 2 × 2 × 1568 = 2 × 2 × 2 × 2 × 2 × 2 × 784 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 392 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 196 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 98 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 49 = 2^10 × 7^2 To find the square root, we take half of each exponent: sqrt(50176) = sqrt(2^10) × 7^2 = 2^10/2 × 7^2/2 = 2^5 × 7^1 = 32 × 7 = 224 The square root of 50176 is 224. b) For 105625:* 105625 = 5 × 21125 = 5 × 5 × 4225 = 5 × 5 × 5 × 845 = 5 × 5 × 5 × 5 × 169 = 5^4 × 13^2 To find the square root, we take half of each exponent: sqrt(105625) = sqrt(5^4 × 13^2) = 5^4/2 × 13^2/2 = 5^2 × 13^1 = 25 × 13 = 325 The square root of 105625 is 325. Step 3: Solve Question 3 by finding the cube number of the given numbers. a) For 22:* The cube number of 22 is 22^3. 22^3 = 22 × 22 × 22 = 484 × 22 = 10648 The cube number of 22 is 10648. b) For 44:* The cube number of 44 is 44^3. 44^3 = 44 × 44 × 44 = 1936 × 44 = 85184 The cube number of 44 is 85184. Step 4: Solve Question 4 by dividing Rs. 100 between Ram and Hari in the ratio 2:3. Total amount = Rs. 100 Ratio of shares for Ram and Hari = 2:3 Sum of the ratio parts = 2 + 3 = 5 Ram's share: Ram's share = (2)/(5) × 100 = 2 × 20 = 40 Hari's share: Hari's share = (3)/(5) × 100 = 3 × 20 = 60 Ram gets Rs. 40 and Hari gets Rs. 60. Step 5: Solve Question 5 by simplifying the expressions. i) (3m)^4 × (3m)^-3 × (3m)^-5(3m)^-2 × (3m)^-3* Using the exponent rule a^x × a^y = a^x+y: = (3m)^4 + (-3) + (-5)(3m)^(-2) + (-3) = (3m)^4 - 3 - 5(3m)^-2 - 3 = (3m)^-4(3m)^-5 Using the exponent rule (a^x)/(a^y) = a^x-y: = (3m)^-4 - (-5) = (3m)^-4 + 5 = (3m)^1 = 3m The simplified expression is 3m. ii) a^a-b × a^b-c × a^c-a* Using the exponent rule a^x × a^y × a^z = a^x+y+z: = a^(a-b) + (b-c) + (c-a) = a^a-b+b-c+c-a = a^0 Using the exponent rule a^0 = 1: = 1 The simplified expression is 1. Step 6: Solve Question 6 by simplifying the expression. $(1)/(4) +